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8/12/2019 Inverse Trigonometry Theory_e
http://slidepdf.com/reader/full/inverse-trigonometry-theorye 1/20
DG^AP
"dgoisafudgrpaysi s.io" 7
Iovlrsl ^ricjojdltri` Huo`tijos
Iotrjnu`tijo 5 ^al stunlot dgy kl hgdibigr gkjut tricjojdltri` huo`tijos viz sio x, `js x, tgo x, `jsl` x, sl` x,`jt x wita rlspl`tivl njdgios [, [, [ ‐ {(;o + 7) /;}, [ ‐ {o }, [ ‐ {(;o + 7) /;}, [ ‐ {o } gon rlspl`tivlrgocls V‐7, 7T, V‐7, 7T, [, [ ‐ (‐7, 7), [ ‐ (‐7, 7), [.
@jrrlspjoniocby, six iovlrsl tricjojdltri` huo`tijos (gbsj `gbbln iovlrsl `ir`ubgr huo`tijos) grl nlhioln.
sio ‐7 x 5 ^al sydkjb sio ‐7 x jr gr`siox nlojtls tal gocbl sj tagt sio 4 x. Gs g nirl`t dlgoioc, sio ‐7 x is ojt ghuo`tijo, gs it njls ojt sgtishy tal rlquirldlots hjr g rubl tj kl`jdl g huo`tijo. Kut ky g suitgkbl `aji`lV‐7, 7T gs its njdgio gon stgongrnizln slt V‐ /;, /;T gs its rgocl, talo rubl sio ‐7 x is g siocbl vgbulnhuo`tijo.
^aus sio ‐7 x is `josinlrln gs g huo`tijo wita njdgio V‐7, 7T gon rgocl V‐ /;, /;T.
^al crgpa jh y 4 sio ‐7 x is gs sajwo klbjw, wai`a is jktgioln ky tgfioc tal dirrjr idgcl, jh tal pjrtijo jh tal
crgpa jh y 4 sio x, hrjd x 4 ‐ /; tj x 4 /;, jo tal biol y 4 x.
`js ‐7 x 5 Ky hjbbjwioc tal nis`ussijos, sidibgr tj gkjvl, wl agvl `js ‐7 x jr gr` js x gs g huo tijo witanjdgio V‐7, 7T gon rgocl V=, T.
^al crgpajh y 4 `js ‐7 x is sidibgrby jktgioln gs tal dirrjr idgcl jh tal pjrtijo jh tal crgpa jh y 4 `js x hrjdx 4 = tj x 4
tg o ‐7 x 5 Sl clt tgo ‐7 x jr gr`tgox gs g huo`tijo wita njdgio [ gon rgocl (‐ /;, /;).Crgpa jh y 4 tgo ‐7 x
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" ;
/;
/;
y
xj
`jsl` ‐7 x 5 `jsl` ‐7 x jr gr``jsl` x is g huo`tijo wita njdgio [ ‐ (‐7, 7) gon rgocl V‐ /;, /;T ‐ {=}.Crgpa jh y 4 `jsl` ‐7 x
sl` ‐7 x 5 sl` ‐7 x jr gr`sl` x is g huo`tijo wita njdgio [ ‐ (‐7, 7) gon rgocl V=, T ‐ { /;}.Crgpa jh y 4 sl` ‐7 x
`jt ‐7 x 5 `jt ‐7 x jr gr``jt x is g huo`tijo wita njdgio [ gon rgocl (=, )Crgpa jh y 4 `jt ‐7 x
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 0
Lxgdpbl # 7 5 Hion tal vgbul jh tgo
0
7tgo
;7
`js 77 .
Pjbutijo 5 tgo
0
7tgo
;7
`js 774 tgo
80 4 tgo
8
4
.
Lxgdpbl # ; 5 Hion njdgio jh sio ‐7 (;x ; ‐ 7)
Pjbutijo 5 Blt y 4 sio ‐7 (;x ; ‐ 7)Hjr y tj kl nlhioln ‐ 7 (;x ; ‐ 7) 7
= ;x ; ; = x ; 7 x V‐7, 7T.
Plbh prg`ti`l prjkblds 5
(7) Hion tal vgbul jh sio
;
7sio
0
7
(;) Hion tal vgbul jh `jsl` Vsl` ‐7 (‐ ; ) + `jt ‐7 ( ‐7)T
(0) Hion tal njdgio jh y 4 sl ‐7 (x ; + 0x + 7)
(>) Hion tal njdgio jh y 4 `js ‐7
;
;
x7
x
(<) Hion tal njdgio jh y 4 tgo ‐7 )7x( ;
Goswlrs 5 (7) 7 (;) ‐7(0) (‐ , ‐ 0T V ‐ ;, ‐ 7T V=, ) (>) [ (<) (‐ , ‐7T V7, )
]rjplrty 7 5 ‒‐x—
^al crgpas jh sio ‐7 x, tgo ‐7 x, `jsl` ‐7 x grl syddltri` gkjut jricio.Alo`l wl clt sio ‐7 (‐x) 4 ‐ sio ‐7 x
tgo ‐7 (‐x) 4 ‐ tgo ‐7 x`jsl` ‐7 (‐x) 4 ‐ `jsl` ‐7 x.
Gbsj tal crgpas jh `js ‐7
x, sl` ‐7
x, `jt ‐7
x grl syddltri` gkjut tal pjiot (=, /;). Hrjd tais, wl clt`js ‐7 (‐x) 4 ‐ `js ‐7 xsl` ‐7 (‐x) 4 ‐ sl` ‐7 x`jt ‐7 (‐x) 4 ‐ `jt ‐7 x.
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" >
]rjplrty ; 5 ^ ‐7
(i) sio (sio 7 x) 4 x, 7 x 7
]rjjh 5 Blt 4 sio ‐7 x. ^alo x V‐7, 7T & V‐ /;, /;T. sio 4 x, ky dlgoioc jh tal sydkjb sio (sio ‐7 x) 4 x
Pidibgr prjjhs `go kl `grriln jut tj jktgio
(ii) `js (`js 7 x) 4 x, 7 x 7(iii) tgo (tgo 7 x) 4 x, x [(iv) `jt (`jt 7 x) 4 x, x [(v) sl` (sl` 7 x) 4 x, x 7, x 7(vi) ` js l ( `js l 7 x) 4 x, |x| 7
^al crgpa jh y 4 sio (sio ‐7 x) `js (`js ‐7 x)
^al crgpa jh y 4 tgo (tgo ‐7 x) `jt (`jt ‐7 x)
^al crgpa jh y 4 `jsl` (`jsl` ‐7 x) sl` (sl` ‐7 x)
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" <
]rjplrty 0 5 ^ ‐7 ^
(i) sio ‐7 (sio x) 4UoT,;/)7o;(,;/)7o;V(x,x)7o;(
T;/o;,;/o;Vx,xo;
]rjjh 5 Ih x V;o ‐ /;, ;o + /;T, talo ‐;o + x V‐ /;, /;T gon sio (‐;o + x) 4 sio x.
Alo`l sio ‐7
(sio x) 4 ‐;o + x hjr x V;o ‐ /;, ;o + /;T.]rjjh jh ; on pgrt is blht hjr tal stunlots.
Crgpa jh y 4 sio ‐7 (sio x)
(ii) `js ‐7 (`js x) 4oT,o;,)7o;V(x,xo;T)7o;(,o;Vx,xo;
Crgpa jh y 4 `js ‐7 (`js x)
y
x
y 4 ;
+ x
p y
4 ‐
x y
4 x y
4 ;
‐ x
p
;
(iii) tgo ‐7 (tgo x) 4 ‐ o + x, o ‐ /; 2 x 2 o + /;, o U
Crgpa jh y 4 tgo ‐7 (tgo x)
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 8
(iv) `jsl` ‐7 (`jsl` x) is sidibgr tj sio ‐7 (sio x)
Crgpa jh y 4 `jsl` ‐7 (`jsl` x)
(v) sl` ‐7 (sl` x) is sidibgr tj `js ‐7 (`js x)
Crgpa jh y 4 sl` ‐7
(sl` x)
(vii) `jt ‐7 (`jt x) 4 ‐o + x, x (o , (o + 7) ), o U
Crgpa jh y 4 `jt ‐7 (`jt x)
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 3
[ldg rf 5 sio (sio ‐7 x), `js (`js ‐7 x), .... `jt (`jt ‐7 x) grl gplrijni` (ojo plrijni`) huo`tijos walrl gssio ‐7 (sio x), ..., `jt ‐7 (`jt x) grl plrijni` huo`tijos.
]rjplrty > 5 ‒7/x—
(i) `jsl` ‐7 (x) 4 sio ‐7 (7/x), |x| 7
]rjjh 5 Blt `jsl` ‐7 x 4 7/x 4 sio sio ‐7 (7/x) 4 sio ‐7 (sio )
4 (gs V‐ /;, /;T ‐ {=})4 `jsl` ‐7 x
(ii) sl` ‐7 x 4 `js ‐7 (7/x), |x | 7
(iii) `jt ‐7 x 4=x),x/7(tgo=x),x/7(tgo
7
7
]rjplrty < 5 ‒ /;—
(i) sio 7 x + `js 7 x 4;
, 7 x 7
]rjjh 5 Blt G 4 sio ‐7 x gon K 4 `js ‐7 x sio G 4 x gon `js K 4 x sio G 4 `js K sio G 4 sio ( /; ‐ K) G 4 /; ‐ K, kl`gusl G gon /; ‐ K V‐ /;, /;T G + K 4 /;.
Pidibgrby, wl `go prjvl
(ii) tgo 7 x + `jt 7 x 4;
, x [
(iii) `jsl` 7 x + sl` 7 x 4;
, x 7
Lxgdpbl # 0 5 Hion tal vgbul jh `jsl`
>0
`jt`jt 7
.
Pjbutijo 5 `jt (`jt ‐7 x) 4 x, x [
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 1
`jt
>0
`jt 7 4>
0
`jsl`
>0
`jt`jt 74 `jsl`
>
04 ; .
Lxgdpbl # > Hion tal vgbul jh tgo ‐7
>0
tgo .
Pjbutijo 5
tgo ‐7 (tgo x) 4 x ih x
;,
;
Gs>
0
;,
; tgo ‐7
>0
tgo>
0
>0
;0
,;
crgpa jh y 4 tgo ‐7 (tgo x) is gs 5
hrjd tal crgpa wl `go sll tagt ih ;
2 x 2;
0,
talo tgo ‐7 (tgo x) 4 x ‐
tgo ‐7
>0
tgo 4>
0 ‐ 4 ‐
>
Lxgdpbl # < 5 Hion tal vgbul jh sio ‐7 (sio3) gon sio ‐7 (sio (‐<)).
Pjbutijo. Blt y 4 sio ‐7 (sio 3)
sio ‐7 (sio 3) 3 gs 3 ;
,;
; 2 3 2;
<
crgpa jh y 4 sio ‐7 (sio x) is gs 5
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 9
Hrjd tal crgpa wl `go sll tagt ih ; x ;
<, talo
y 4 sio ‐7 (sio x ) `go kl writtlo gs 5y 4 x ‐ ;
sio ‐7 (sio 3) 4 3 ‐ ;
Pidibgrby ih wl agvl tj hion sio ‐7 (sio(‐<)) talo
‐ ; 2 ‐ < 2 ‐;
0
hrjd tal crgpa jh sio ‐7
(sio x), wl `go sgy tagtsio ‐7 (sio(‐<)) 4 ; + (‐<) 4 ; ‐ <
Lxgdpbl # 8 5 Hion tal vgbul jh js ‐7 {sio( ‐ <)}
Pjbutijo 5 Blt y 4 `js ‐7 {sio(‐<)}4 `js ‐7 (‐ sio <)4 ‐ `js ‐7 (sio <) (`js ‐7 (‐ x) 4 ‐ `js ‐7 x, |x| 7)
4 ‐ `js ‐7
<;
`js ..........(i)
Ojtl tagt 5 ‐ ; 2
<;
2 ‐
crgpa jh `js ‐7 (`js x ) is gs 5
Hrjd tal crgpa wl `go sll tagt ih ‐ ; x ‐ talo `js ‐7 (`jsx) 4 x + ;
hrjd tal crgpa `js ‐7
<;
`js 4
<; + ; 4
<;
<
hrjd (i), wl clt
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 7=
y 4 ‐
<;
< y 4 < ‐
;0
.
Lxgdpbl # 3 5 Hion tal vgbul jh tgo
0;
`jt 7
Pjbutijo 5 Blt y 4 tgo
0;
`jt 7 ........(i)
`jt ‐7 (‐x) 4 ‐ `jt ‐7 x, x [(i) `go kl writtlo gs
y 4 tgo
0;
`jt 7
y 4 ‐ tgo
0
;`jt 7
`jt ‐7 x 4 tgo ‐7
x
ih x ? =
y 4 ‐ tgo
;0
tgo 7 y 4 ‐;0
Lxgdpbl # 1 5 Hion tal vgbul jh sio
>0
tgo 7 .
Pjbutijo 5 sio
>0
tgo 7 4 sio
<0
sio 7 4<0
Lxgdpbl # 9 5 Hion tal vgbul jh tgo
0<
`js;7 7
Pjbutijo 5 Blt y 4 tgo
0<
`js;7 7 ..........(i)
Blt `js ‐7
0<
4
;,= gon `js 4
0<
(i) kl jdls y 4 tgo
;
..........(ii)
tgo ;
;4
`js7`js7
4
0
<7
0<
74
<0
<04
>)<0( ;
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 77
tgo;
4 ·
;
<0.........(iii)
;
>,= tgo
;? =
hrjd (iii), wl clt y 4 tgo;
4
;<0
Lxgdpbl # 7= 5 Hion tal vgbul jh `js (;`js ‐7 x + sio ‐7 x) walo x 4<7
Pjbutijo 5 `js
<7
sio<7
`js; 77 4 `js
<7
`js<7
sio<7
`js 777
4 `js
<7
`js;
74 ‐ sio
<7
`js 7 .........(i)
4 ‐;
<7
7
4 ‐<
8;.
G bi t l r 5 Blt<7
`js 7 4 `js 4<7
gon
;,=
sio 4<;>
sio ‐7 (sio ) 4 sio ‐7
<;>
..........(ii)
;,= sio ‐7 (sio ) 4
lqugtijo (ii) `go kl writtlo gs
4 sio ‐7
<;> 4 `js ‐7
<7
`js ‐7
<7
4 sio ‐7
<;>
Ojw lqugtijo (i) `go kl writtlo gs
y 4 ‐ sio
<;>
sio 7 ........(iii)
<;>
V‐7, 7T sio
<;>
sio 74
<;>
hrjd lqugtijo (iii), wl clt y 4 ‐<;>
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 7 ;
Lxgdpbl # 77 5 Pjbvl sio ‐7 (x ; ‐ ;x + 7) + `js ‐7 (x ; ‐ x) 4;
Pjbutijo 5 sio ‐7 (h(x)) + `js ‐7 (c(x)) 4;
h(x) 4 c(x) gon ‐7 h(x), c(x) 7
x; ‐ ;x + 7 4 x ; ‐ x x 4 7, g``lptln gs g sjbutijo
Plbh prg`ti`l prjkblds 5
(8) Hion tal vgbul jh `js
8siosio 7
(3) Hion tal vgbul jh sio
>0
`js`js 7
(1) Hion tal vgbul jh js ‐7 (`js 70)
(9) Hion sio ‐7 (sio ), `js ‐7 (`js ), tgo ‐7 (tgo ), `jt ‐7 (`jt ) hjr
0,;
<
(7=) Hion tal vgbul jh `js ‐7 (‐ js >) (77) Hion tal vgbul jh tgo ‐7
13
tgo
(7;) Hion tal vgbul jh tgo ‐7
>7
`jt (70) Hion tal vgbul jh sl
0;
`js 7
(7>) Hion tal vgbul jh `jsl
0
7sio 7
(7<) Hion tal vgbul jh sio (; js ‐7 x + sio ‐7 x) walo x 4<7
(78) Pjbvl tal hjbbjwioc lqugtijos (i) < tgo ‐7 x + 0 `jt ‐7 x 4 ; (ii) > sio ‐7 x 4 ‐ `js ‐7 x
(73) Lvgbugtl tgo
>>7
l``js 7 (71) Lvgbugtl sl
8078
`jt 7
(79) Lvgbugtl sio
>0
`jt;7 7 (;=) Lvgbugtl tgo
><7
tgo; 7
(;7) Pjbvl sio ‐7 (x ; ‐ ;x + 0) + `js ‐7 (x ; ‐ x) 4;
Goswlrs 5 (8);0
(3) ojt nlhioln
(1) 70 ‐ > (9) 0 ‐ , ‐ ; , ‐ 0 , ‐ ;
(7=) > ‐ (77)1
(7;)
;>7
(70);0
(7>) ‐ 0 (7<)<7
(78). (i) x 4 7 (ii) x 4 ;7
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 70
(73)<>
(71)788<
(79)<
<;(;=)
733
(;7) Oj sjbutijo
]rjplrty 8 5 I nlotit ils jo gn ni tij o g on su ktrg `ti jo 5
(i) sio 7 x + sio 7 y
7yx&=y,=x,x7yy7xsio
7)yx(&=y,=x,x7yy7xsio
;;;;7
;;;;7
]rjjh 5 Blt G 4 sio ‐7 x gon K 4 sio ‐7 y walrl x, y V=, 7T.
sio (G + K) 4 x ;y7 + y ;x7
sio ‐7 sio (G + K) 4 sio ‐7
;; x7yy7x
sio ‐7
;; x7yy7x
4K G;/hjr )K G(
;/K G=hjr K G4
7yx),ysiox(sio7yx,ysioxsio
;;77
;;77
(ii) sio ‐7 x ‐ sio ‐7 y 4 sio ‐7
;; x7yy7x 6 x, y V=, 7T
(iii) `js ‐7 x + `js ‐7 y 4 `js ‐7 ;; y7x7xy 6 x, y V=, 7T
(iv) `js ‐7 x ‐ `js ‐7 y 4
7xy=6y7x7xy`js
7yx=6y7x7xy`js
;;7
;;7
(v) tgo ‐7 x + tgo ‐7 y 4
7xy&=y,xih xy7yx
tgo
7xy&=y,xih xy7yx
tgo
7xy&=y,xih ;/7xy&=y,xih ;/
7
7
(vi) tgo ‐7 x ‐ tgo ‐7 y 4 tgo ‐7
xy7yx
, x =, y =
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DG^AP
"dgoisafudgrpaysi s.io" 7>
Ojtls 5 (i) x; + y; 7 & x, y = = s io 7 x + sio 7 y
;
gon x ; + y; 7 & x, y =
; sio 7 x + sio 7 y
(ii) xy 2 7 gon x, y = = tgo 7 x + tgo 7 y 2;
6 xy ? 7 gon x, y = ;
2 tgo 7 x + tgo 7 y 2
(iii) Hjr x 2 = jr y 2 = talsl inlotitils `go kl usln wita tal albp jh prjplrty ‒‐ x—i.l. `agocl x jr y tj x jr y wai`a grl pjsitivl.
Lxgdpbl # 7; 5 Pajw tagt sio ‐7
<0
+ sio ‐7
737<
4 ‐ sio ‐7
1<1>
Pjbutijo 5 <0
? =,737<
? = gon;
<0
+;
737<
43;;<1;;8
? 7
sio ‐7
<0
+ sio ‐7
737<
4 ‐ sio ‐7
;<9
7737<
;19;;<
7<0
4 ‐ sio ‐7
<>
.737<
731
.<0
4 ‐ sio ‐7
1<1>
Lxgdpbl # 70 5 Lvgbugtl `js ‐7
707;
+ sio ‐7
<>
‐ tgo ‐7
7880
Pjbutijo 5 Blt z 4 `js ‐7
707;
+ sio ‐7
<>
‐ tgo ‐7
7880
sio ‐7
<>
4;
‐ `js ‐7
<>
z 4 `js ‐7
707;
+
<>
`js;
7 ‐ tgo ‐7
7880
.
z 4;
‐
707;
`js<>
`js 77 ‐ tgo ‐7
7880
.........(i)
<>
? =,707;
? = gon<>
2707;
`js ‐7
<>
‐ `js ‐7
707;
4 `js ‐77897>>
7;<78
7707;
<>
4 `js ‐7
8<80
lqugtijo (i) `go kl writtlo gs
8/12/2019 Inverse Trigonometry Theory_e
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DG^AP
"dgoisafudgrpaysi s.io" 7<
z 4;
‐ `js ‐7
8<80
‐ tgo ‐7
7880
z 4 sio ‐7
8<80
‐ tgo ‐7
7880
.........(ii)
sio ‐7 8<80 4 tgo ‐7 7880
hrjd lqugtijo (ii), wl clt
z 4 tgo ‐7
7880
‐ tgo ‐7
7880
z 4 =
Lxgdpbl # 7> 5 Lvgbugtl tgo ‐7 9 + tgo ‐7
><
Pjbutijo 5 9 ? =,>< ? = gon
><9 ? 7
tgo ‐7 9 + tgo ‐7
><
4 + tgo ‐7
><
.97
><
94 + tgo ‐7 (‐ 7) 4 ‐
>4
>
.
Lxgdpbl # 7< 5 Nlhiol y 4 `js ‐7 (>x 0 ‐ 0x) io tlrds jh `js ‐7 x gon gbsj nrgw its crgpa.
Pjbutijo 5 ]grt - 75 Blt y 4 js ‐7 (>x0 ‐ 0x) Njdgio 5 V‐7, 7T gon rgocl 5 V=, T
Blt `js ‐7 x 4 V=, T gon x 4 js y 4 `js ‐7 (> `js 0 ‐ 0 `js )
y 4 `js ‐7 (`js 0 ) ..... .... ..( i)
Hic.5 Crgpa jh `js ‐7 (`js x) V=, T 0 V=, 0 T tj nlhiol y 4 `js ‐7 (`js 0 ), wl `josinlr tal crgpa jh `js ‐7 (`js x)
io tal iotlrvgb V=, 0 T. Ojw, hrjd tal gkjvl crgpa wl `go sll tagt
(i) ih = 0 `js ‐7 (`js 0 ) 4 0 hrjd lqugtijo (i), wl clt
y 4 0 ih 0
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DG^AP
"dgoisafudgrpaysi s.io" 78
y 4 0 ih = 0
y 4 0 `js ‐7 x ih ;7
x 7
(ii) ih 0 ; `js ‐7 (`js 0 ) 4 ; ‐ 0 hrjd lqugtijo (i), wl clt
y 4 ; ‐ 0 ih 0 ;
y 4 ; ‐ 0 ih 0
2 0
;
y 4 ; ‐ 0`js ‐7 x ih ‐ ;7
x 2;7
(iii) 0 0 `js ‐7 (`js 0 ) 4 ‐ ; + 0 hrjd lqugtijo (i), wl clt y 4 ‐ ; + 0 ih 0 0
y 4 ‐ ; + 0 ih 0
;
y 4 ‐ ; + 0 `js ‐7 x ih ‐ 7 x ‐ ;7
hrjd (i), (ii) & (iii), wl clt
y 4 `js ‐7 (>x 0 ‐ 0x) 4
;7
x76x`js0;;
7x
;
76x`js0;
7x;7
6x`js0
7
7
7
]grt-; 5 Hjr y 4 `js ‐7 (>x 0 ‐ 0x)njdgio 5 V‐7, 7Trgocl 5 V=, T
(i) ih ;7
x 7 , y 4 0 `js ‐7 x.
nxny
4 ;x7
04 ‐ 0(7 ‐ x ; ) ‐7/; ...........(i)
nxny
2 = ih x
7,;7
nl`rlgsioc ih x
7,;7
gcgio ih wl nihhlrlotigtl lqugtijo (i) w.r.t. ‑x’, wl clt
;
;
nx
yn4 ‐ ;/0; )x7(
x0
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DG^AP
"dgoisafudgrpaysi s.io" 73
;
;
nx
yn2 = ih x
7,;7
`jo`gvity njwowgrns ih x
7,;7
(ii) ih ‐;7
x 2;7
, y 4 ; ‐ 0`js ‐7 x.
nxny
4 ;x7
0
nxny
? = ih x
;7
,;7
io`rlgsioc ih x
;7
,;7
gon ;
;
nx
yn4 ;/0; )x7(
x0
(g) ih x
=,;7
talo ;
;
nx
yn2 =
`jo`gvity njwowgrns ih x
=,;7
(k) ih x
;7
,= talo ;
;
nx
yn? =
`jo`gvity upwgrns ih x
;7
,=
(iii) Pidibgrby ih ‐ 7 x 2 ‐;7
talonxny
2 = gon ;
;
nx
yn? =.
tal crgpa jh y 4 `js ‐7 (>x 0 ‐ 0x) is gs
Plbh prg`ti`l prjkblds5
(;;) Lvgbugtl sio ‐7
<>
+ sio ‐7
70<
+ sio ‐7
8<78
(;0) Ih tgo ‐7 > + tgo ‐7 < 4 `jt ‐7 , talo hion ‑ ’
(;>) ]rjvl tagt ; `js ‐7
70
0+ `jt ‐7
8078
+;7
`js ‐7
;<3
4
(;<) Pjbvl tal lqugtijo tgo ‐7 (;x) + tgo ‐7 (0x) 4>
(;8) Pjbvl tal lqugtijo sio ‐7 x + sio ‐7 ;x 40
;
(;3) Nlhiol y 4 sio ‐7 (0x ‐ >x 0) io tlrds jh sio ‐7 x gon gbsj nrgw its crgpa.
(;1) Nlhiol y 4 tgo ‐7
;
0
x07
xx0io tlrds jh tgo ‐7 x gon gbsj nrgw its crgpa.
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DG^AP
"dgoisafudgrpaysi s.io" 71
Goswlrs. (;;);
(;0) 4 ‐9
79(;<) x 4
87
(;8) x 4;7
(;3) y 4 sio ‐7 (0x ‐ >x 0) 4
;7x76xsio0
7x;7
6xsio0
;7
x;7
6xsio0
7
7
7
crgpa jh y 4 sio ‐7 (0x ‐ >x 0)
(;1) y 4 tgo ‐7
;
0
x07
xx04
x0
76xtgo0
0
7x6xtgo0
0
7x
0
76xtgo0
7
7
7
Hic.5 Crgpa jh y 4 tgo ‐7
;
0
x07
xx0
] rjpl rty 3 5 Dis `lbbgol jus inlo ti ti ls
(i) sio 7
;x7x; 4
;7
;7
;7
x7ih xsio;
7xih xsio;
|x|ih xsio;
7
7
7
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DG^AP
"dgoisafudgrpaysi s.io" 79
crgpa jh y 4 sio 7
;x7x;
(ii) `js 7 (; x ; 7) 4=x7ih x`js;;7x=ih x`js;
7
7
crgpa jh y 4 `js 7 (; x ; 7)
(iii) tgo 7;x7
x; 4
7xih xtgo;7xih xtgo;7|x|ih xtgo;
7
7
7
crgpa jh y 4 tgo 7;x7
x;
(iv) sio 7 ;
7 ;
x
x 4
7xih xtgo;7xih xtgo;7|x|ih xtgo;
7
7
7
crgpa jh y 4 sio 7 ;
7 ;
x
x
(v) `js 7;
;
x7
x74
=xih xtgo;=xih xtgo;
7
7
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DG^AP
crgpa jh y 4 `js 7;
;
x7
x7
(vi) Ih tgo 7 x + tgo 7 y + tgo 7 z 4 , talo x + y + z 4 xyz
(vii) Ih tgo 7 x + tgo 7 y + tgo 7 z 4;
, talo xy + yz + zx 4 7
(viii) tgo 7 7 + tgo 7 ; + tgo 7 0 4
(ix) tgo 7 7 + tgo 7
;7 + tgo 707 4 ;